Extension of a conditional performance score for sample size recalculation rules to the setting of binary endpoints

Background Sample size calculation is a central aspect in planning of clinical trials. The sample size is calculated based on parameter assumptions, like the treatment effect and the endpoint’s variance. A fundamental problem of this approach is that the true distribution parameters are not known before the trial. Hence, sample size calculation always contains a certain degree of uncertainty, leading to the risk of underpowering or oversizing a trial. One way to cope with this uncertainty are adaptive designs. Adaptive designs allow to adjust the sample size during an interim analysis. There is a large number of such recalculation rules to choose from. To guide the choice of a suitable adaptive design with sample size recalculation, previous literature suggests a conditional performance score for studies with a normally distributed endpoint. However, binary endpoints are also frequently applied in clinical trials and the application of the conditional performance score to binary endpoints is not yet investigated. Methods We extend the theory of the conditional performance score to binary endpoints by suggesting a related one-dimensional score parametrization. We moreover perform a simulation study to evaluate the operational characteristics and to illustrate application. Results We find that the score definition can be extended without modification to the case of binary endpoints. We represent the score results by a single distribution parameter, and therefore derive a single effect measure, which contains the difference in proportions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{I}-p_{C}$$\end{document}pI-pC between the intervention and the control group, as well as the endpoint proportion \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{C}$$\end{document}pC in the control group. Conclusions This research extends the theory of the conditional performance score to binary endpoints and demonstrates its application in practice. Supplementary Information The online version contains supplementary material available at 10.1186/s12874-024-02150-4.

In the following, we derive the distribution of the normal approximation test, given in Equation (3).Here, a right arrow with d or p on top denotes convergence in distribution and probability respectively.
For the numerator of the test statistic, it follows from the central limit theorem that For the denominator, it follows from the law of large numbers XI + XC Since the denominator converges against a constant, we can derive the asymptotic distribution of the normal approximation test statistic using slutsky's theorem following, we present the derivation using which leads to the simplified representation Our goal is to find a parameter λ, which parameterizes the mean and variance component of the test statistic's distribution simultaneously.The parameter Hence, and therefore Plugging this result together with the definition of .Since the combined test statistic is given by the conditional power corresponds to From Equation (4) it follows that Hence, it holds that

C.2 Derivation of the recalculated sample size formulas
The basic idea of the recalculation rules considered in the simulation is to calculate the sample size ñ, such that the observed conditional power reaches a predefined targeted value CP λ(z 1 , ñ) = 1 − β.Note that the formula for the observed conditional power corresponds to the formula for the true conditional power, with the true parameter λ replaced by the estimated parameter λ.
[3] This leads to Note that in the above equation, ñ is not necessarily an integer.Therefore, the result needs to be rounded up to get the recalculated sample size.
From Formula (C9) for ñ and Formula (C7) for CP λ(z 1 , n), the formulas of all the considered recalculation rules can be derived.To simplify the formulas we define N max rec := n max − n 1 .Below, we provide the formulas for the observed conditional power approach, the restricted observed conditional power approach, and the promising zone approach.Remember that N rec refers to the second stage sample is sufficient to characterize the endpoint's distribution.)Since N rec is stochastic in our setting, we only need to consider the type I error rate for various combinations of n 1 and p C .The results are given in Figure S1.The results were generated by the simulation described in Section 4.2.The type I error rate corresponds to a simulation of the true global power for a standardized treatment effect of λ = 0.
We observe that for small values of p C , our designs tend to have an error rate

)
Our goal was to derive a representation of the asymptotic distribution of the normal approximation test, based on a single parameter.Note that Wassmer and Brannath[2] provide a related idea.They derive a single distribution parameter ζ, which parameterizes the power of the normal approximation test.The definition of ζ by Wassmer and Brannath[2] differs from the definition of the parameter λ in this paper.However, both are strongly related as they can represent concepts derived from the asymptotic test statistic distribution (like the power) by a single distribution parameter.Appendix C: Adaptive designs for binary endpoints C.1 Derivation of the conditional power We set w1 :=

below 2 .
5%.Only for large values of p C exceeding of the type I error rate occurs.For large values of n 1 , the designs barely exceed the specified type I error rate, which is why we can conclude that the type I error rate does not pose a problem in our considered main setting with n 1 = 50 and p C = 0.3.Note, that the relatively small type I error rate for the restricted OCP is due to the additional option to stop for futility if CP λ(Z 1 , n max ) is below the specified threshold.

Figure
Figure S1 type I error rate for different values of n 1 and pc.

E. 2
Power, mean sample size, and conditional performance scoreIn the following, we want to check whether, in the finite sample case, the single parameter λ is sufficient to parameterize the (global) power, mean sample size, and conditional performance score of the considered recalculation rules.Therefore, we check whether the performance measures, visualized as a function of λ, behave differently for different values of p C .If so, it is not sufficient to present results only depending on λ and a differentiated representation for different values of p C is necessary.To limit the scope of the analysis, we keep n 1 = 50 fixed.We observe in FiguresS2, S3, S4, S5 and S6 that differences for different values of p C are rather marginal.Accordingly, the relationship between λ and the mean sample size, the power as well as the conditional performance score does almost not depend on p C .Accordingly, the 1-dimensional parameterization of mean sample size, power, and conditional performance score by λ is approximately correct for the considered setting.

Figure S2
Figure S2 Performance measures for pc = 0.02 and different values of λ